Overview of the `gyrotropic` module¶

The gyrotropic module of postw90 is called by setting gyrotropic = true and choosing one or more of the available options for gyrotropic_task. The module computes the quantities, studied in ¹, where more details may be found.

`gyrotropic_task=-d0`: the Berry curvature dipole¶

The traceless dimensionless tensor

\[ \begin{equation} \label{eq:D_ab} D_{ab}=\int[d{\bm k}]\sum_n \frac{\partial E_n}{\partial{k_a}} \Omega_n^b \left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n}, \end{equation} \]

`gyrotropic_task=-dw`: the finite-frequency generalization of the Berry curvature dipole¶

\[ \begin{equation} \label{eq:D-tilde} \widetilde{D}_{ab}(\omega)=\int[d{\bm k}]\sum_n \frac{\partial E_n}{\partial{k_a}}\widetilde\Omega^b_n(\omega) \left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n}, \end{equation} \]

where \(\widetilde{\bm\Omega}_{{\bm k}n}(\omega)\) is a finite-frequency generalization of the Berry curvature:

\[ \begin{equation} \label{eq:curv-w} \widetilde{\bm\Omega}_{{\bm k}n}(\omega)=- \sum_m\,\frac{\omega^2_{{\bm k}mn}}{\omega^2_{{\bm k}mn}-\omega^2} {\rm Im}\left({\bm A}_{{\bm k}nm}\times{\bm A}_{{\bm k}mn}\right) \end{equation} \]

Contrary to the Berry curvature, the divergence of \(\tilde{\bm\Omega}_{{\bm k}n}(\omega)\) is generally nonzero. As a result, \(\widetilde{D}(\omega)\) can have a nonzero trace at finite frequencies, \(\tilde{D}_\|\neq-2\tilde{D}_\perp\) in Te.

`gyrotropic_task=-C`: the ohmic conductivity¶

In the constant relaxation-time approximation the ohmic conductivity is expressed as \(\sigma_{ab}=(2\pi e\tau/\hbar)C_{ab}\), with

\[ \begin{equation} \label{eq:C_ab} C_{ab}=\frac{e}{h}\int[d{\bm k}]\sum_n\, \frac{\partial E_n}{\partial{k_a}} \frac{\partial E_n}{\partial{k_b}} \left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n} \end{equation} \]

a positive quantity with units of surface current density (A/cm).

`gyrotropic_task=-K`: the kinetic magnetoelectric effect (kME)¶

A microscopic theory of the intrinsic kME effect in bulk crystals was recently developed ²³.

The response is described by

\[ \begin{equation} \label{eq:K_ab} K_{ab}=\int[d{\bm k}]\sum_n\frac{\partial E_n}{\partial{k_a}} m_n^b \left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n}, \end{equation} \]

which has the same form as Eq. \(\eqref{eq:D_ab}\), but with the Berry curvature replaced by the intrinsic magnetic moment \({\bm m}_{{\bm k}n}\) of the Bloch electrons, which has the spin and orbital components given by ⁴

\[ \begin{equation} \label{eq:m-spin} \begin{aligned} m^{\rm spin}_{{\bm k}n}&=-\frac{1}{2}g_s\mu_{\rm B} \langle\psi_{{\bm k} n}\vert\bf \sigma\vert\psi_{{\bm k}n}\rangle\\ % \label{eq:m-orb} {\bm m}^{\rm orb}_{{\bm k}n}&=\frac{e}{2\hbar}{\rm Im} \langle{\bm\partial}_{\bm k}u_{{\bm k}n}\vert\times (H_{\bm k}-E_{{\bm k}n})\vert{\bm\partial}_{\bm k}u_{{\bm k}n}\rangle, \end{aligned} \end{equation} \]

where \(g_s\approx 2\) and we chose \(e>0\).

`gyrotropic_task=-dos`: the density of states¶

The density of states is calculated with the same width and type of smearing, as the other properties of the gyrotropic module

`gyrotropic_task=-noa`: the interband contributionto the natural optical activity¶

Natural optical rotatory power is given by ⁵

\[ \begin{equation} \label{eq:rho-c} \rho_0(\omega)=\frac{\omega^2}{2c^2}{\rm Re}\,\gamma_{xyz}(\omega). \end{equation} \]

for light propagating ling the main symmetry axis of a crystal \(z\). Here \(\gamma_{xyz}(\omega)\) is an anti-symmetric (in \(xy\)) tensor with units of length, which has both inter- and intraband contributions.

Following Ref. ⁶ for the interband contribution we writewe write, with \(\partial_c\equiv\partial/\partial k_c\),

\[ \begin{equation} \label{eq:gamma-inter} \begin{gathered} {\rm Re}\,\gamma_{abc}^{\mathrm{inter}}(\omega)=\frac{e^2}{\varepsilon_0\hbar^2} \int[d{\bm k}] \sum_{n,l}^{o,e}\, \Bigl[ \frac{1}{\omega_{ln}^2-\omega^2} {\rm Re}\left(A_{ln}^bB_{nl}^{ac}-A_{ln}^aB_{nl}^{bc}\right) \\ -\frac{3\omega_{ln}^2-\omega^2}{(\omega_{ln}^2-\omega^2)^2} \partial_c(E_l+E_n){\rm Im}\left(A_{nl}^aA_{ln}^b\right) \Bigr]. \end{gathered} \end{equation} \]

The summations over \(n\) and \(l\) span the occupied (\(o\)) and empty (\(e\)) bands respectively, \(\omega_{ln}=(E_l-E_n)/\hbar\), and \({\bm A}_{ln}({\bm k})\) is given by Berry Eq. [Berry connection matrix].

Finally, the matrix \(B_{nl}^{ac}\) has both orbital and spin contributions given by

\[ \begin{equation} \label{eq:B-ac-orb} B_{nl}^{ac\,({\rm orb})}= \langle u_n\vert(\partial_aH)\vert\partial_c u_l\rangle -\langle\partial_c u_n\vert(\partial_aH)\vert u_l\rangle \end{equation} \]

and

\[ \begin{equation} \label{eq:B-ac-spin} B_{nl}^{ac\,({\rm spin})}=-\frac{i\hbar^2}{m_e}\epsilon_{abc} \langle u_n\vert\sigma_b\vert u_l\rangle. \end{equation} \]

The spin matrix elements contribute less than 0.5% of the total \(\rho_0^{\rm inter}\) of Te. Expanding \(H=\sum_m \vert u_m\rangle E_m \langle u_m\vert\) we obtain for the orbital matrix elements

\[ \begin{equation} \label{eq:Bnl-sum} B_{nl}^{ac\,({\rm orb})}=-i\partial_a(E_n+E_l)A_{nl}^c \sum_m \Bigl\{ (E_n-E_m) A_{nm}^aA_{ml}^c -(E_l-E_m) A_{nm}^cA_{ml}^a \Bigr\}. \end{equation} \]

This reduces the calculation of \(B^{\text{(orb)}}\) to the evaluation of band gradients and off-diagonal elements of the Berry connection matrix. Both operations can be carried out efficiently in a Wannier-function basis following Ref. ⁷.

`gyrotropic_task=-spin`: compute also the spin component of NOA and KME¶

Unless this task is specified, only the orbital contributions are calcuated in NOA and KME, thus contributions from Eqs. \(\eqref{eq:m-spin}\) and \(\eqref{eq:B-ac-spin}\) are omitted.

S. S. Tsirkin, P. Aguado Puente, and I. Souza. Gyrotropic effects in trigonal tellurium studied from first principles. ArXiv e-prints, October 2017. arXiv:1710.03204. ↩
T. Yoda, T. Yokoyama, and S. Murakami. Current-induced orbital and spin magnetizations in crystals with helical structure. Sci. Rep., 5:12024, 2015. URL: http://dx.doi.org/10.1038/srep12024, doi:10.1038/srep12024. ↩
S. Zhong, J. E. Moore, and I. Souza. Gyrotropic magnetic effect and the magnetic moment on the fermi surface. Phys. Rev. Lett., 116:077201, Feb 2016. URL: https://link.aps.org/doi/10.1103/PhysRevLett.116.077201, doi:10.1103/PhysRevLett.116.077201. ↩
Di Xiao, Ming-Che Chang, and Qian Niu. Berry phase effects on electronic properties. Rev. Mod. Phys., 82:1959–2007, Jul 2010. doi:10.1103/RevModPhys.82.1959. ↩
E.L. Ivchenko and G.E. Pikus. Natural optical activity of semiconductors (tellurium). Sov. Phys. Solid State, 16(7):1261, 1975. URL: https://www.scopus.com/inward/record.uri?eid=2-s2.0-0016444392&partnerID=40&md5=d44204b27eb4356a166f389a0f8c8a4e. ↩
A. Malashevich and I. Souza. Band theory of spatial dispersion in magnetoelectrics. Phys. Rev. B, 82:245118, 2010. doi:10.1103/PhysRevB.82.245118. ↩
J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza. Spectral and fermi surface properties from wannier interpolation. Phys. Rev. B, 75:195121, 2007. ↩

Overview of the gyrotropic module¶

gyrotropic_task=-d0: the Berry curvature dipole¶

gyrotropic_task=-dw: the finite-frequency generalization of the Berry curvature dipole¶

gyrotropic_task=-C: the ohmic conductivity¶

gyrotropic_task=-K: the kinetic magnetoelectric effect (kME)¶

gyrotropic_task=-dos: the density of states¶

gyrotropic_task=-noa: the interband contributionto the natural optical activity¶

gyrotropic_task=-spin: compute also the spin component of NOA and KME¶